2
Quiz#
KEY
Name:
Prove theorem: T-6.3 (product of any two orthogonal matrices is orthogonal)
1.
(ABthB) =
dd,{s = B-tB *.7
are orthonormal)
Prove theorem T-6.4 (Columns or rows of an orthogonal matrix
2.
[ro"l
Ls
t ?l
lnor- '(v,t''ltwe i
CtTa, - brb -- CTc =
cJb- *rc= |{ct 4076* Lrc*cr(=a
Pgorrry
|
a,ue6,ui,ytrf {-tAn'F
a rE, o
*^ilul"
ontLu^'o**Q M's
U
3.
Prove theorem T-6.5 (det(Rproper)
d,"b
s.^1yos€
iroa*fro$r* r
det(Rimproper)
: -1)
2
(Rtg = d,/f[Rt) /rtt(g) = Ai (&)'= d-'l(4
rd,rt[R)=
(o f *n-:
t\
:-rI,
!\
R-=
a,{r,
[a\.6tt]
t
c u**l'*'uw"^--A ueg'hP
*W{t"l'6ta*t])=
: *f
1**$1)
*
dd{fot6ld't I ta tr" I s xal)
I
=
Page 2
4.
Prove theorem T-6.8 (If R is orthogonal 3 by 3 matrix then
(HINT: suppose (Ra)x(Rb):
(R'lx(R4)=
R(axb),
-R
(Ra)- = RdRr
is proper orthogonal)
((1*{)
4tut,M ?^*U/-r%
= RGd)
ea)(na1
of1
4
re
ffiln = Ra
ffi* Raf
5
.
[-
Prove that rot(k,
*
&im{?)
=
ft,*
,I +
sin(p)
E + (1
fi-cdDY)
-
cos(
g17
8'
is a proper orthogonal matrix
fflrl- e
:
71o
s 4,r 71 "
Je
6.
cp)
Prove
?"'k' =" &'
+ rl,n^(r) Wfti ,, -c^@)i'/e
er
e irr^ce L ** +l ro* f e,v) il f rof
tr(rot(k,9)
:
I+2cos(rP)
air*(v)fr *
t"{roCfi,,p}+r I t u
' ] =*
5?"t*))'fr
f t **f*)) *.te&r*
r 3 + t **t*) Frfd-
: i :,:*?;?'lggi
7.
Derive rotation matrix R
F= rd+fes, y)=
if
k=
e3, (p
:60u
lr^,r.-Qrru,f
r) =
-"f
:t*; =,
,l
[t -E
+ z a'^rc)
o1
?rll= l9 L ?l
i"*v ehY
e
L0
L;;
ffi
CS656,Qtiz#2
8.
Paee 3
Given are vectors
b
by
450:
s
:
s:lI,4,2lr
to*
(*,
and
b:
[0,3,01t . Find vector c which results from rotation of a about
,., t)q,
Find ft and
cp
,d=
Iqt
l;
f =r*'f(e*,',s)o'=
,
9.
[3] hr fil= u'
Elrrr
c, llq l= I'{-l
o
I
Ett.l L+
bEO
which correspond to the rotation
oer'(f)= 36
o $lz -!?
Lo tlz '1312
o =l
-R --
o 1
a+,tn\Y = ?6
--cr^Y | yo+(u,v)
=
l" Yn
wl
to
'nn\
t
? k= lrl=
10.
[
matrix
0
[l e
8l
l.Ar/
a
-t^
1=3U
Find the rotation matrix which corresponds to rotation about x-axis by 600, followed by rotation
about z-axis by 900
lot[?r,q,
rot (t,, no)
:
? -E
?]
L l=
-to][r
[o o?il,
i;
[o o rll.ut
of7
L]
-i
io os-l
lr o
lu e i
I
I
CS656, Quiz #2
0.8910
T_
ttr
6B
I 1. Given is a homogeneous transform matrix:
0
01
0.4540
0
-0.4s40-
1
0
3
0.8910
0
Find its inverse (HINT: Consider the rotation part of Z orthogonal)
T=.
I {n^q Tir s- HJ:
r'Ir -f
r rl
ft
l-:r|
t-'
I
F= Gl
'--r
Ld\\l 'l
f$il:$rq- foT''
[;r I JI
; :U:l
i
*Yfti:
i--o*Y?,o.
o t>
t
$ lt
;$]:il]
;
o D I
[jl,
l-D
t
l2.FindHTmahixwhichmovJvectoragivingvector
b(b=Th),whereaandbaredefinedasinfigure
4l
fr
&=
lr'
f4r
r(h,r){=
L
x
b
T
ds
450
t:
I'l
q lEj
2l
6\tre I
3l
L;J
t
Write the operator equation first, then expand the matrices, then multiply the
' = Trc*rrnQ,,-
[r o 0 -1 I
lv I o o LE
T:
I;Bt
w rite
R"+
@t,-4s")")
I
€:
e"
Ie
lz
EJ)
2
Lo
2_
dO
{-a
the screw matrix for x-axis with rotation 300 and translation 10 mm
ft"t (r,,i{
1; q'u) (3,,t) =
\ o O rol
o E- -i 0
o! I e
o o a lJ
|
I
,.\
-'{
|
",1=[l-tr trt2 ol
I2lo ol
il Lf :tilt
I
fuelql la-- ls e s'Jt
|
t3.
a)
.
!:,i:
H
Lo
,-T
I ro et
J---t
ll
rl
'|
ffi
l
Page 6
CS656, Quiz #2
14. Given is vector
v
:
[5 6 Tlr.Express v in terms of basis vectors
{:5 ei+
€1, €2
illld
of7
€3'
GQy+1ez
15. Given are fivo matrices:
o .l
[r o
R,=rot(u,a)=10 0.9397
lo
os4zo
-0.3420
|
e848 ol
0
0'1736
o, = rot(v,p)=lo'sa+t
|
fo.tzro
[ o
o.%sl)
-0
o
l.l
(a) Find vectors u andv, and angles aand B'
tA
= Br Cr4t4 = a.q3q7
l\f; €g
errfs) =
6'tv?b
o{= Zoo
f1
= 8o"
i
(b) Suppose that the two matrices represent basis vectors of two coordinate systems, { 1} ald {2} , :
' ' respectively. Given is vector t*:1L2,0.75,4.01r defined in {1}, find the same vector'x defined in
rx,
.R1 an R2), then compute the numeric
(in terms of
1Zi. Writeihe solution in generic iorm first
solution (you may use MATLAB for this).
\
R,tx =
ri'
R.?X
tx= nln\ x
W
PageT of7
C5656, Quiz #2
16.
system by 300, then about z-axis by
Suppose an object rotates about x-axis of the World coordinate
60".
(a) What would be the equivalent angle of rotation?
po{ [ee,
Tt=
C,rrY =
to) bt (e,,3 o) =
:
i]
f+
fr -E;u 'l[
,i
tl
utl'F-:l=\+
f
il
lA
t; o rJL"i EJ [o L Vl
+'tR)-l_* t*€.-€-J
2-
tf=.
&
hf/vJ
=
3{3.3=
v,'Eqs
-r'
8
(o,3cas)= 1,tffir cod= 60'45'
(b) Would the equivalent angle of rotation be the same if the two rotations took place in reversed
order fiustify your answer)?
*l&o' ouu€
1l- roul.{ {^-n*Lq {t { * B) = ft (BA )
/a-r
JL*,k,n
:
h (ro't(er,6o) rot( g,,ir) :
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